Is it possible to export a multiplayer game from BGE?
Short answer:  No, at least not without a lot of effort.

Long answer:

The engine has no baked-in network multiplayer.  You have to set up the whole thing yourself with Python scripting.  It's all quite complex.

As far as I know this is the most in-depth explanation of the subject:
http://youtu.be/4xZRfzOtxzA

^ Should you decide to build that into a full game, be sure to swap pickle for something like json.  Using pickle on untrusted data is a nightmarish fiery hell of insecurity.

If you're familiar with Python and compiling, check out the Python bindings for ENet:
https://github.com/aresch/pyenet

At one point someone was working on an addon to ease the burden for common folk, but last I checked it wasn't matured enough for actual use.  The project might be dead as well:
http://blenderartists.org/forum/showthread.php?242740-Blender-Game-Engine-Multiplayer-Add-on